The logic of negationless mathematics

C OMPOSITIO M ATHEMATICA

P. G. J. V REDENDUIN

The logic of negationless mathematics

Compositio Mathematica, tome 11 (1953), p. 204-270

<http://www.numdam.org/item?id=CM_1953__11__204_0>

© Foundation Compositio Mathematica, 1953, tous droits réservés.

L’accès aux archives de la revue « Compositio Mathematica » (http:

//http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utili- sation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

by

P. G. J. Vredenduin

Arnhem

Griss stated a new method of

treating

intuitionistic mathematics without

using negation 1).

The intention of this paper is ^to

give

a

corresponding logical system.

1. We first list the

syntactical

rules and afterwards

give

^their

interpretation.

The

following signs

^{are used:}

1. atomic formulas

F(x), F(x, y), F(x,

y,

z),

^...,

G(x),

^...,

=

(x, y), ^# (x, y);

^x, y, z, ... ^are called

variables; they

^are sup-

posed

^{to be}

^different,

2. undefined

signs

^{A, v,}

(Ev), (v), ~vs (v

stands for a

variable,

vs for a sequence of different

variables).

Definition

by

induction of ^a

zvell-lormed formula (wff):

a. every atomic formula is ^a

wff,

b.

if p and q

^are

wff,

then p ^A q and _p v q ^are

wff,

c. if p

and q

^are

wff, then p

~vs q is ^a

wff,

d. if p is ^a

wff,

^then

(Ev)p

^and

(v )p

^{are wff.}

Definition

by

induction of

free

and bound variables:

a. any variable

occurring

^{in an} atomic formula is free in that

formula,

b. any variable that is free

(bound) in p,

^{is free}

(bound) in p^q

and

in p

^v q,

any variable that is free

(bound)

in q, is free

(bound) in p

A q and in p ^v q,

1) ^{G. F. C.} Gitiss, Negatieloze intuïtionistische wiskunde, ^{Versl. Ned. Akad.}

v. Wetensch., ^afd. Natuurk., ^LIII (1944), p. 261-268,

Negationless intuitionistic mathematics, Verh. Kon. Ned. Akad. v. W’etenseh., XLIX (1946), p. 1127-1133, ^LIII (1950), p. 4562013463.

Logique ^des mathématiques intuitionistes sans négation, Comptes ^{rendus des}

séances de l’Acad. des Sc., ^{t. 227} (1948), p. 946-948.

c. any variable

belonging

^{to vs that} is free in p

(or q),

^{is bound}

in p ~vs q, and is said to be bound

by ~vs,

d. if the variable v is free in p, it is bound in

(Ev)p

^{and in}

(v)p,

and is said to be bound

by (Ev)

^and

(v), respectively,

e. any variable that is free in p

(or q)

^{and not bound}

by

~vs,

by ( Ev)

^or

by (v),

^{is free} in p ~vs q,

(Ev)p, (v)p, respectively.

An

arbitrary

wff will be written p, q, ^{r, ....} If we want to ex-

press, that the wff p

contains

the free

variable x,

^{we write}

p(x).

This

only

^means that p contains the free variable x, but not that

x is the

only

variable that is free in p.

There is no difference _{between p} and

p(x) occurring

^{in the same}

derivation.

p(x)

^{is written at those}

places where

it is essential to

remember that x occurs free in p; ^at other

places

^{of the same}

derivation _{p may be} written.

If in a derivation

first,

e.g.,

p(x)

and afterwards

p(y)

^occurs,

then with

p(y)

^{is meant the}

^wff,

^{that is}

generated

^from

p(x) by replacing

every x, that is free in

p(x) by

y.

If _x, y, z, ^{... are} all the free variables of p, then

3p

^{stands for}

(Ex)(Ey)(Ez) ...

p. If ^no variable is free in p, then

3p

^{stands for}

p. The order of the

(Ex), (Ey), (Ez),

^... is indifferent. This will be shown afterwards

(7.21).

If vs is the sequence of variables x, y, z,

..., then 3V8P

^{stands for}

(Ex)(Ey)(Ez) ...

p, and

(vg)p

^for

(x)(y)(z) ...

p.

p, q, r, ^{... are} not

signs belonging

^{to the}

system. They

^are

merely

^{names for}

arbitrary

^{wff. So}

they belong

^{to the}

metasystem.

The

signs

^v8,

3, 3V8’ (vs),

~vs

belong

^{also to the}

metasystem.

Interpretation.

^The

sign

^A is used for

conjunction,

^{v for dis-}

junction, (Ev)

is the existentional

operator, (v)

^the

all-operator.

Wff without free variables are to be

interpreted

^as

propositions,

wff with free variables as

propositional

^functions.

p(x)

-z

q(x)

^means that the class determined

by

^the

proposi-

tional function

p(x) (short:

^{the class}

p(x))

is included in the class

q(x). 1) p(x, y)

~xy

q(x, y)

^means that the class

(of pairs

^x,

y) p(x, y)

is included in the class

q(x, y). p(x, y)

^-z

q(x, y)

^{is to be}

interpreted

^as the class of those y for which

p(x, y)

is included in

q(x, y),

^etc.

(Ex)p(x)

^{is a}

proposition; (Ex)p(x, y)

is the class of those y for which an x exists that satisfies

p(x, y).

^{The same} holds for the

all-operator.

1) ^{It is} supposed ^{in this} and the next two paragraphs, ^{that p} and q ^{contain no}

free variables different from those mentioned between brackets.

There is still some

difficulty

^{with the}

interpretation

of e.g.

p(x )

~xy

q(x, y). If p(x)

^{was a class of}

^x’s,

^{it could not be seen in}

what

respect

this class

might

be included in

q(x, y).

^{But we} may also

interpret p(x)

^as the class of

pairs

^x, y that

satisfy p(x).

^So

if

p(x)

^{as a}

propositional

function of _x, is satisfied

by

^a, it is satis- fied

by

^any

pair

^a, y. This situation is

analogous

^to

solving

^the

equations

^{x + 1 = 0 and 3x +}

5y

^{= 2.} The solution is x ^{= -} 1, y ⁼ 1, ^{as x = - 1} and y

arbitrary

^will

satisfy x

^{+ 1 = 0. So}

there is the same kind of

ambiguity

^{in the}

interpretation

of pro-

positional

^{functions as in the}

meaning

^{of an}

equation.

In the same way it is

possible

^to

interpret

^a wff without free variables as a

propositional

function. A wff without free variables that is true may be

interpreted

^{as an}

all-class,

^an all-class of

pairs,

etc., ^{and is to be}

compared

^{with an identical}

equation.

^{This kind}

of

interpretation

^{enables us in formal}

respect

^{not to} discern any

more between

propositions

^and

propositional

functions. We can

restrict ourselves to theorems about

arbitrary

wff that may ^or may ^not contain free variables.

It is clear from the

foregoing that p

n q is to be

iiiterpreted

^as

the

product

^{of the}

classes p

and q, p v q ^as their sum.

The

negationless

method. Before

continuing

it will be necessary to

explain

ⁱⁿ brief the fundamental ideas of Griss’ method.

Griss

accepts

^{that in}

constructing

^a mathematical

system

^we

progress from ^true

propositions

^{to other}

propositions

^{that are also}

true.

Perhaps

^we may, when

making

^a

rough calculation,

^{find the}

impossibility

^{that some} theorem will ever be a

part

^{of our}

system.

That result _may be _very instructive for the

investigator,

^{but it}

is not a

part

^{of the}

system

itself. When 1 am

building

^{a house} it may be of

great importance

^to decide that 1 shall not use a certain kind of

bricks,

^but this decision does not make those bricks

part

^{of the}

house. So in the mathematical

system only

^those

propositions

^will

occur that are true. And as these

propositions

^are all affirmative

(the contradictory propositions being only possible

ⁱⁿ

rough"),

^a

sign

^for

negation

is useless in his

system.

Another fundamental feature of Griss’ method is that he

accepts

that in

constructing

^{one is}

always constructing something

^{and so}

never will construct

nothing.

In accordance with this view he declares that the null-class does not exist.

Every propositional

function has the

property

^{that it can} be satisfied. So the

product

of two classes is not

always

^a class. If the classes have ^no lement in common their

product

^{is not the}

null-class,

^but

merely

^senseless.

The

propositional

calculus. In a mathematical

Griss-system only

true

propositions

^{will occur.} So there is no reason for

linking

^them

by

^a

sign

^for

disjunction

^or

implication. "In rough"

^we may find that ^a certain

proposition

^{can be}

proved

as soon as A has been

proved

^{and also} as soon as B has been

proved.

^{And then we}

might

say that A ^v B

implies

^{C. But in the}

system

^{itself we shall never}

progress from A

(or

^from

B )

^{to another}

proposition

^{before A}

(or B)

^{has been}

proved.

So in the

system

itself the

disjunction

^of

propositions

is useless. The same holds for the

implication.

^In

rough

^we may convince ourselves that B ^can be

proved

^{as soon as}

A has been

proved,

but this consideration is not a

part

^{of the}

system

^itself.

Formally linkimg propositions by

^{v or ~ is}

possible,

but the

interpretation

of the result is the same as the

interpreta- ’

tion of their

conjunction (this

remark is of

importance,

^{as we shall}

formally

^treat

propositions

^and

propositional

functions in the

same

way).

Propositions

may be linked

by conjunction.

As the kind of

linking obeys

^{the same laws as the}

linking

^of

propositional

^func-

tions,

^{there is no reason for a}

separate propositional

^calculus.

Axioms and derivations.

There are two kinds of axioms:

a. axioms of the form

b. axioms of the form: if

Definition

by

^{y induction}

of - (p is

^derivable

^{rom P).}

p

(p

^{zs erzvable}

f

In this definition

P, Q, Q1, Q2

^{stand for}

arbitrary

finite sequences of wff.

1. If every q, that

belongs

^to

Q, belongs

^to

^P,

^and

if Q

^{is an}

axiom,

^then

-.

p

2.

If Q ’

^{is an} axiom and every q, that

belongs

^to

Q,

^{is derivable}

fr o m

P,

^,

then P.

p

3. If

a. _{Q1 q,}

b.

if Q1,

^then

^Q2,,

^{is an}

^axiom,

q p

c. every q, that

belongs

^to

Q2 , is

^{derivable from}

^P,

then

-.

p

Semantical 1°elnark.

The

meaning

^{of a} dérivation of t he form

p(x, y, ...) q(x, y, ...) _{is, that}

_for

arbitrary

values of _{x, y,} ^...

q(x,

y,

...)

^{can be derived from}

p(x,

y,

...).

So there is a close connection between the derivation of one

propositional

function from another and the inclusion of the classes determined

by

^{the two functions.}

The use

of

^dots.

We discern left and

right

dots. Left dots stand to the left of ^a letter or of -,

right

^{dots to the}

right.

The scope of ^a left

(right) complex

of dots is extended to the left

(right)

^{until a}

right (left) complex

of dots is

reached,

that consists of an

equal

^{or a}

larger

number of

dots,

or, if this is not the _{case, to} the end of the formula.

A and v bind

stronger

^than ~vs.

Final remarks.

In

principle logical

^{theorems can be}

dispensed

with. Their pur- pose is

merely

^to enable abbreviations in the mathematical pro-

cess. Instead of a

large quantity

^of

applications

^{of the}

logical

axioms one

application

^{of a}

logical

theorem may be used.

The mathematician will

perhaps

say that he is ^not

reasoning

ⁱⁿ

detail

according

^{to the}

logical

axioms. But the

logician only

says that it is

possible

^to rebuild the mathematical

system by using

^the

logical

axioms. As soon as it turns out that his

logical system

^is

unable to describe the mathematical

system,

^the

logical system

should be altered. On the other hand the considerations of the

logician

may be of ^some influence on mathematical

thought.

Investigating

^the

logical system

^it will appear that it

obeys

^its

own rules and axioms. But as its structure is very

simple, only

few of its axioms are sufficient for its own foundation. This last remark has a

metalogical

^{character and will not be}

analyzed

further.

We now start

building

^the

logical system.

Axioms will be marked

A,

definitions D and theorems without ^a letter. At the end of the bar the numbers of the

(main) ^axioms,

definitions and theorems

are mentioned that are used.

Definitions are

merely

^{used as} abbreviations.

I. The functional calculus without

considering

^the

inner , structure of the wff 2. The axioms

of conjunction.

A2.0 does not mean that any ^two

propositional

^functions

(classes)

^{have a}

product.

^{We must not}

forget

^{that an axiom can}

only

^{then be}

applied,

^{when the}

premisses

^are derived formulas.

So the

meaning

will be: any x

(or

any

pair

^x, y,

etc.)

^{that satisfies}

p and q, will also

satisfy

p ^A q.

In case p and q ^are

propositions

^A2.0

simply

says that ^two de-

rived

propositions

may be

conjuncted..

Proof.

2.1

Prool.

2.2

Prool.

3. Axiom-s about _~vs avd

~vs.

No variable of vs must be free in r. The

premiss

^r may be

dropped.

The

~vs-operator

^{is of extreme}

importance

ⁱⁿ

negationless logic.

E.g.,

ⁱⁿ

ordinary logic

^{no one} would hesitate to

accept p ~vsq p^r ~vs q.

But in

negationless logic

this derivation is

only possible,

^{if it is}

known that p ^{A r} exists. Therefore the

premiss ~vs.

p ^{A r} has to be added. From this

example

^{it is seen} that in many ^cases additional

premisses

of the form

~vs

p will

distinguish

^the

present

^calculus

from the usual

logical

^calculi.

A3.2 states that any wff

(class)

^that

previously

occurs as a

conclusion,

exists. For

repeated application

of this axiom leads to

3p

^{and to}

~vs

p.

A3.3 states that, ^if

previously

it has been

proved that p(x,

y,

... )

is included in

q(x,

y,

... ),

then there is a sequence x, y, ^... that satisfies

p(x, y, ... )

^{and also a} sequence that satisfies

q(x,

y,

... ).

It is not clear that in A3.4 the

premiss 3 vs

p ^must be added. For if in a mathematical

system

this axiom is

applied,

p is the ^con- clusion of a

preceding

derivation and so the condition

~vs

p will

always

be fulfilled. Still we are not in accordance with the in- tention of our

system,

^if

~vs

p is cancelled. For

according

^{to 2.2}

r

^.

_{Canceling ~vsp}

^we would find

r p~vsp.

And then A3.3 would

give

^the conclusion

3118

p. The derivation of the existence of an

arbitrary

p from ^an

arbitrary premiss

^{r is}

certainly

^{not in}

accordance with our aim.

Proof.

Proo f .

Prool.

Remark. The full

proof

^is:

So p ^{A r turns} out not to be a

premiss

of the derivation of p ^{A r} ~vsq from p ~vs q and

3V8

pr. This is the

meaning

of the asteric in the above

proof.

Proof.

Proo f .

Proof.

1) 3", pq ^{is short for} 3t.,. p A q.

Proof.

Prool.

Proof.

In the 2nd derivation vs’ is the sequence of free variables _{of p.}

We have still to derive

3q

^from

3V8’

q. In case q contains ^a free

variable,

^{that does not}

belong

^to

vs’,

^we

apply

^{A3.2. In case a}

variable of vs’ is not free

in q,

^we

apply

^a theorem that will be

proved

afterwards

(7.00).

3.300

If p r q

^{and no free variable of r is} free in p, then

Proof.

^Similar.

Further 3.300.

Proof.

P.roof.

Prool.

Proof.

Proof.

Prool.

4.

Disjunction.

In A4.0 p v q ^can

only

^be derived from p,

if q

^{is a}

propositional

function. So the

premiss 3q

^{has to be added.}

In A4.4 p ^{A r.} v . q ^{A r can}

only

^be

derived,

^{if the}

products

p ^{1B r}

and q

^{^ r exist.}

Proof. A4.1,

3.30, ^A4.2.

Proof. A4.0,

^3.300

and,

^if necessary, A3.2 and 7.00.

Proof. A3.2,

^{A4.2 or 4.00.}

4.10

if p and q , ^{then v}

r r r

Proof. vs

is the sequence of free variables of p and q.

1) A3.2 should be applied, ^{in case} p and q ^{do not} contain the same free variables.

Proo f .

Proof, vs

^{is the} sequence ^{of free} variables of p.

analogous

Proof.

Further

Proof.

We first prove

3q ^q, by applying (if necessary)

^A3.2

and 7.00

(cf.

^the

proof

^of

3.30).

^Further

A4.0,

^A3.4.

vs contains all free variables of p.

1) ^A3.2 should be applied, if p ^{^.} p ^v q contains ^a free variable that does not

belong ^{to vs.}

Prool.

Proof.

Proo f .

Prool.

analogous

Further

Proof.

analogous

analogous

Further

(1), (2),

^4.11.

Prool.

^4.36.

Proof.

^vs is the sequence of free variables of p, q and ^r.

analogous

analogous

analogous

Further

Proof.

Further

(3), (4), (5), (6),

^{3.301, 4.100.}

Remark. With the _square brackets in the 2nd and the 6th line of the

proof

^{is meant,} that the addition of the

premisses 3pq,

p,

3qr and q

^{v r,}

3pr, 3qr

^{is not} wanted for the derivation there but afterwards for

using

^4.100.

II. The

général

functional calculus

5. We shall ^now introduce all- and

existentional-operators.

p ^{must not} contain x as a free

variable; p

may be

dropped.

p ^{must not} contain x ^{as a free}

variable;

p may be

dropped.

Proof.

Proof.

Proof.

Further

Proof.

Further

Proo f .

Further

6. Rules

of

substitution.

if

p q(x)

and p does ^not contain x as a free

variable,

^then

p ,

^if

^- and q

^{does not contain x as a free}

variable,

q y)

q then

p(y).

q

p(x)

^and

q(x)

^{must not}

contain y

^{as a} bound variable.

p(y)

^and

q(y)

^are formed from

p(x)

^and

q(x) by

^substi-

tuting

y for x ^at every

place

where x is free.

A6.1 If x is bound in p, y does ^{not occur} in p and p is transformed into

p’ by substituting

y for the bound variable x at every

place

^{where it occurs}

(including

ⁱⁿ

the

binding operators), _p

^then

p .

Proof.

7. A7 If x does not occur as a free variable in p, then

(Ex)p (x)p.

If x does not occur as a free variable in p, the theorems ^7.00- 7.03 hold.

If q

^{and r do not contain x as a free variable and}

Proof.

^{5.0, 7.00.}

7.11 If p does ^not contain x as a free variable

and p q, then p (x)q.

Proot.

7.20

Proof.

Further 7.21

Proof.

Further 7.22

Proo f .

Further 7.3

Prool.

^{If vs contains}

variables,

^{that are not free in} p,

they

^can

be

dropped (7.00).

Further A3.2.

8.

hrtplication..

There is some difference in

meaning

^between

p -.,,q and

(x) .

p-,,q.

In both cases p is a

part

^of q. But in the second _case, p is ^a

part

^of

q

for

any ^x. That is

only possible,

^{if for} any ^x p exists,

i.e., (x)(Ey)p.

Proof.

Proof.

Further

Proo f .

y ^{must not} be bound in p ^or q.

Proof. A3.0,

^A6.0.

The theorem is

proved

^{in the same} way for

propositional

^func-

tions

containing

^{more than one variable.}

y ^{must not} be bound in

p(x)

^or

r(x).

Proof.

If y is bound in

q(x), by

^A6.1

q(x)

^{can be} transformed into

q’(x)

^not

containing

y ^{as a} bound variable.

Prool.

8.32 If p does ^not contain y ^{as a} free

variable,

^then

1) Or, ^{if x} is not free in q, p9x) ~xq

p(y) q.

2) ^{It is allowed} that y is free in q(x); ^{tlis will} become clear in section 10.

Proof.

8.33

If q

^{does not} contain y ^{as a} free

variable,

^then

Proof.

8.34 If x is not free in p

and q,

^then

p ~vsq p ~vs’ q.

vs does not contain _x; vs’

consists

of vs and x.

Proof.

Proof.

8.3, ^A3.4.

9. The ba,yic relations = and

#.

It is

possible

^to

apply

^the

logical theory

^{to a field of} individuals.

We presuppose that the individuals ^are discernable. In ^{case we} want to express that ^two individuals are

discernable,

^{we write}

x

# y,

^{in case}

they

^are identical x ⁼ y. The relations = and

#

are introduced as basic relations of our

logical system by

^means

of the axioms A9.020133.

x = y and x # y ^are atomic formulas

(cf. D9.020131).

The use of the

propositional

^{function x #} y renders it

impossible

that there is

only

^one

individual,

or, ^more

precise,

^{makes it}

necessary that there ^{are at} least two discernable individuals. So after

adjunction

^{of the}

sign

#, ^this

theory

^{cannot be}

applied

^to

a field that consists of

only

^one individual.

Formally

this circumstance

might

^be

expressed by

^{the axiom}

(Ex)(Ey)x # y.

But this axiom is not an axiom similar to the

others, but,

^one

might

say, ^a material axiom

(as

^it supposes ^a

special property

of the scope of tlie field of

individuals). Adding

a material axiom

implies adding

material theorems. Instead of

splitting

the theorems in two différent

kinds,

^we

prefer writing

^the

theorems that presuppose the

axiom" (Ex)(Ey)x # y,

^{in the}

usual _way,

(Ex)(Ey)x # y p.

^{But we} shall omit the

premiss (Ex)(Ey)x # y

ⁱⁿ the formulation of

theorems, except

^{in case it}

is the

only premiss.

Following CTriss 1)

^{we choose as axioms:}

p ^{must not} contain _y as a bound variable.

Proof.

1) Vcrsl. Ned. Akad. v. ’Vetensch., ^afd. Natuurk., ^{L I II} (1944), p. 262 and ^266.

. 2) ^{From an} intuitionistic point of view this axiom is suspect; cf. section 15.

Griss proves that it is valid for real numbers.

Proof.

Proo f .

Proof.

^{9.1, 6.02.}

Prool.

Prool.

Proof.

Proof.

10.

Disfunction.

This axiom _says: if x

belongs

^{to the sumclass}

of p

and q, but is different

(discernable)

^from all the members of the class p, then x

belongs

^to q.

Or,

ⁱⁿ

ordinary language,

^{if x}

belongs

^to the sumclass

of p

and q, but ^{not to} p, it

belongs

^to q. But in the last ^sentence it is

negated

^{that x}

belongs

^to p,

perhaps because p(x)

^{turns out}

to be

contradictory.

The former sentence is free from

negation,

because it

only

says that all the members of the

class p are

^different

from x.

Perhaps

^the

following example

makes the difference clearer. 1

am

looking

for my

fountain-pen.

¹ ask: "Is it on my

writing-table?"

1 find it in _my

pocket.

^{And now 1} say: "It is not on my

writing- table,

for it is in _my

pocket."

^{That is a}

negated

^{sentence. But 1}

can also

investigate

every

object

^on my

writing-table

^and

always

find: this

object

is different from my

fountain-pen.

^{Then all the}

objects

^{on the table are} different from my

fountain-pen.

^{And if}

1 know in some way that my

fountain-pen belongs

^{to the sum-}

class of the

objects

^on my table and in my

pocket,

^{I am able to}

conclude

(A10) :

my

fountain-pen

^{is in} my

pocket.

D10 If x is the

only

free variable

of p(x),

^then

-P(x) = dfp(y)~y u # x.

If x

and y

^{are the}

only

free variables

of p(x, y),

^then

~p(x, y ) = dfp(u, v ) ~uv u # x v # y,

^etc.

By

this definition a kind of

negation

is introduced. But this

operation,

^~, is based upon the relation of difference. So it is ^not

a

negation

in the proper sense, ^as it has

nothing

^to do with refu- tation or contradiction.

Still, formally,

^{it has} many

properties

ⁱⁿ

common with the usual

negation.

We are now able to formulate A10 in a

simpler

^form:

A10.0

p v q ""p

q

This axiom is more

general,

^as the number of variables is arbi-

trary.

Rernark. There is still some

ambiguity

^with

respect

^{to the}

-"négation"

^of

propositional

functions with variables that have

been identified.

E.g.,

^{if in}

~p(x, y )

the variables are

identified,

^we

get according

to D10

But if we consider

p(x, x )

^{as a}

propositional

function with one

variable,

then D10 says

We choose the first définition. This is done

by

^the

following

^de-

cision : if a variable in ^a

propositional

function is

repeated,

^the

function should

formally

be treated as a function of two

(or more)

identified variables and not as a function of one variable. So identification of variables does not reduce the number of variables.

~x # x,

considered as a function of one

variable,

^{would be}

nonsense as the

dass x # x

^is

empty. So x #

^x would be senseless and cannot be

negated.

~x # x,

considered as a function of two identified variables

means u #

^v ~uv u # x ^v

v #

x, and this is

significant.

^{It can}

be derived from 3 #

by

A9.2 and A3.4.

(Cf. 10.12.)

Under certain existentional conditions there is no harm in

negating

^a function of two variables in the same way ^{as a} function of one variable. This will be shown in 10.10201311.

We define:

~p(x, x, z)

= af

(Ex)(Ey)(Ez)p(x, y, z),

^etc.

So if the

3-operator

^is

applied

^{to a wff} with identified free

variables,

the variables should first be

changed

into different variables and then

they

all should be bound

by (Evs).

The definitions of ^~ and 3

applied

^{to wff} with identified free variables have the

following

consequence. If ^a theorem has been

proved

for wff without identified free

variables,

^the

corresponding

theorem for wff with identified free variables is an immediate consequence of it

(by

^{means of}

A6.0).

Mind that the

3-premisses

remain

unchanged,

when free variables are identified in the pre- misses and the conclusion of ^a derivation. So in

proofs

^{we are}

always

^{allowed to} suppose that all free variables ^are different.

Remark. It seems that

by

^the

following

derivation we are able to construct a

disjunction

^{of two wff of which one}

represents

^an

empty

^class.

But this wff should not be

interpreted

^{as the sum} of the class

f(x)

and the

empty class y ^#

y. We first form the class of

triples (x,

y,

.z),

^that

satisfy f(x )

v y

# z

and from this class we form the subclass of those

triples

of which the second and third element

are identical. So ^we find as

interpretation

^{the class of}

triples (x, y, y)

of whieh ir satisfies

f(x)

^and y is

arbitrary.

Proof. Suppose

^that p contains

just

^two free variables.

In case p contains ^more than two free

variables,

^the

proof

^is

similar.

More

generally

^we prove in the same way:

10.100 If xo, xl, ..., Xn, y0, y1,..., ym is the sequence of free variables of _p, then

Prool. Suppose that p

^contains

just

^two free variables.

Further 7.11, 8.0, ^D10.

In case p contains ^more than two free

variables,

^the

proof

^is

similar.

More

generally

^we prove in the ^same way :

10.110 If _{x0, x1, ..., Xng} y0, y1,...,ym is the sequence of free variables of p, then

Proof. Suppose that p

^contains

just

^{two free variables.}

The

proof

^is

analogous

^{in case} p contains ^more than two free variables.

10.12 shows that from the

negation

^of

p(x, x),

considered as a

function of tzvo

variables,

^{can be} derived the

negation

^of

p(x, x),

considered as a function of one

variable,

^but

only

^{if the}

premiss (Ex)p(x, x)

is valid. If this

premiss

^{were not}

^valid,

^{the conclusion}

would be senseless.

More

generally

^we prove in the ^same way:

10.120 If xo, xl, ^..., xn, y0, yi, ^..., Ym is the sequence of free variables of _p, then

Proof.

10.21

Proof.

10.22

Proo f .

10.23

Proof.

10.40

Proof. Suppose that p

^contains

only

^one free variable.

If the number of free variables of p is ^more than _one, the

proof

is similar.

Prool. Suppose x

^{is the}

only

^free variable of _p and of q.

The

proof

^is

analogous,

if p

and q

^{contain more than one, but}

the same free variables.

Suppose

^{that p}

and q

^{do not} contain the same free variables.

E.g.,

^{the free variables} of p are ^x

and y,

of q ^x and z. Then we

define

Now first we

prove -, p’ ^(9.1).

^Therefore

~q’ ~p’ ~p’.

And from this we

prove

10.50 remains

valid,

^{if a}

premiss

^is

^added,

^{that does not contain}

a free variable that is free in p ^or q.

Prool.

^Similar.

Proof. Suppose x

^{is the}

only

^free variable of p and of q.

Further similar to the

proof

^{of 10.50.}

10.52 remains

valid,

^{if a}

premiss

^is

^added,

^{that does not contain}

a free variable that is free in p ^or q.

Prooi. A3.3,

10.42, 10.51, 10.43, ^3.20.

Proof.

Further 3.301, ^4.100.

Prool.

Let p

and q

^{contain one free}

^variable;

the free variable of _p is the same as the free variable of q.

Further ^a similar derivation of

~q(x),

^{and A2.0.}

The

proof

^is

analogous

in tlie other cases

(cf.

^the

proof

^of

10.50).

Proo f .

Proo f .

Let r be the

only

free variable _{of p} and of q.

The

proof

is similar

if p and q

^{contain more or} different free variables.

Proof.

This theorem is not in conflict with intuitionism. It

merely

shows that in

negationless

intuitionistic mathematics _y = x ~ z=x can

only

^be

"negated"

^{in those cases in which}

y # x

^{v z} # x ^can be

proved.

^{So the}

possibilities of "negating"

^{in this} system ^{are more} restricted than in normal intuitionism.

11. Individual constants.

In the

application

^{of the}

theory

it may ^be

possible,

that indivi- dual constants are substituted for variables. For this reason and for other _reasons, that will appear

later,

^we

enlarge

^{the used}

signs

with

3. individual constants a,

b,

^....

As a

metasystematical symbol

^{for an}

arbitrary

individual con-

stant, ^{we shall use the letter c.}

To the definition of a well-formed formula we add:

e.

if p(x)

^{is a wff and c an} individual constant,

and p(r)

^is

changed into q by replacing

every x, that is free

in p(x),

by

^c,

then q

^{is a wff.}

The new wff q is written

p(c).

The

following

^{axioms are added.}

11.0 If x does not occur as a free variable

in q,

^and

p(x) q,

then

p(c)

.

q

Proof. Al l .1,

^7.10.

11.1 If x does not occur as a free variable in q,

and P

then

p (q(c).

Proof.

^{7.11, All.O.}

If a wff

containing

^an individual constant is

negated,

^{the con-}

stant is treated in the same way ^{as a} free variable.

So,

e.g.,

~P(c) = dfp(x) ~x x # c,

~p(x,

^c,

c) =df p(y, z, u) ~yzu y # x ~ z # c ~ u # c.

A constant

occurring

twice is treated in the same way ^{as two} identified free variables.

If an

3-operator

^is

applied

^{on a wff}

containing

^{an individual}

constant, ^{the constant has to be}

replaced by

^{a free variable.}

So, e.g.,

3P(C)

= df

3p(x),

~p(x,

^c,

c)

=af

~p(x,

y,

z).

This has the

following

consequence. If ^a theorem has been

proved

^{for wff not}

containing

individual constants, ^the

correspond- ring

theorem for wff

containing

individual constants is an imme- diate consequence of it

(by

^{means of}

A11.2).

Note that the 3-

premisses

^remain

unchanged,

when free variables are

replaced by

individual constants in the

premisses

and the conclusion of a

dérivation. So in

proofs

^{we are}

always

^{allowed to} suppose that the wff do not contain individual constants.

Remark. _{If CI}

#

C2, then

f(x)

^v CI ⁼ C2 is ^to be

interpreted

^as

the class of

triples (x,

Cl,

C2),

^{of which x satisfies}

f(x).

Definition. If p and q ,

^we say that p and q ^are

equivalent,

q p

and write _{p ~ q.}

The relation ^~ is a

metasystematical

^relation.

11.2 If p ~ q and ^{r =-= s,} then

a. p ~ r ~ q ~ s

(A2.1, A2.0)

b. p ~ r ~ q ~ s

(4.2)

c. p ~vs r ~ q ~vs s

(A3.0, A3.3,

3.30,

A3.4)

d.

3p

⁼

3q (3.30)

e.

(Ex)p

^~

(Ex)q (5.0) f. (x)p

^~

(x)q (A5.1)

g.

p(c)

^-

q(c) (Ail.2).

From 11.2 it is seen

by

^induction

that, i f p and q ,

p and q ^are

interchangeable.

^{q p}

Semantical remarks.

Suppose

that x is the

only

free variable of

p(x)

^{and of}

q(x).

^We remember, ^that

p(x)

^v

q(x)

^{is a}

propositional

^function

determining

the sumclass of the classes determined

by p(x)

^and

q(x).

^So

p(c)

^v

q(c)

^{will mean, that c is a} member of this class. Therefore

p(c)

^v

q(c)

^{is not a}

disjunction

^{of the}

propositions p(c)

^and

q(c).

In case

only

^{one of}

p(c)

^and

q(c)

^{is true,}

p(c)

^and

q(c)

^{would not}

both be a

proposition

^{and the}

disjunction p(c)

^v

q(c),

if understood

as a

disjunction

^of

propositions,

would be senseless. But

p(c)

^v

q(c)

understood as one

proposition

^{and not}

composed

^{out of two} pro-

positions

^{is not} senseless and

merely

^{means, that c}

belongs

^{to the}

sumclass

p(x)

^v

q(x).

There is another

difficulty. Suppose

^{that x}

and y

^{are the}

only

free variables of

p(x, y)

^and

q(x, y).

^{How is}

p(x, c)

^v

q(x, c)

^{to be}

understood?

Again

^{it does not mean the}

disjunction of p(x, c)

^and

q(x, c).

For it is

possible,

that there exists no x for

which,

_e.g.,

p(x, c)

holds. And then

p(x, c)

^{is not a}

propositional

function. So

we should not be able to form

p(x, c)

^v

q(x, c)

^as soon as one of the two does not

represent

^a class that is not

empty.

Therefore we choose a different

interpretation,

^{that is}

closely

connected with the

interpretation of p(c)

^v

q(c).

^The

propositional

function

p(x, y)

^v

q(x, y)

determines a class of

pairs (x, y).

^Now

we

decide,

^that

p(x, c)

^v

q(x, c)

determines the subclass of those

pairs

^of

which y

is the individual constant c. This

interpretation

^is

independent

of the existence of the functions

p(x, c)

^and

q(x, c)

separately.

E.g.,

^x

^#

^{0 in the}

theory

of whole numbers is

equivalent

^with

x 0 ^{v x} &#x3E; 0. The

proposition

¹ # ^{0 is true.} Therefore the

proposition

1 0 ~ 1 &#x3E; 0 is true too,

though

^{1 0}

separately

is not a

proposition.

And | x | # 1 y 1

^is

equivalent

^{with X2 C}

y2

^{v x2} &#x3E;

y2.

^So

x2 ₀ v x2 &#x3E; 0 determines the class of those x that are # ^{0. But} x2 0

separately

^{is not a}

propositional

^function.

This causes some difficulties. The

meaning

^{of x2 0 v x2} &#x3E; 0

depends

^{on 0}

being

^{or not}

being

^obtained

by

substitution in a

preceding

formula. In the former case it is

sensible,

in the latter senseless. We

decide,

that every individual ^constant

appearing explicitly

^{in a formula is}

supposed

^to be introduced

by

substitution for a free variable. In the next section we shall see how it will be

possible

to construct

propositional

functions in which individual constants occur

implicitly

^{that are not}

supposed

^{to be}

introduced by

substitution.

12. Note about

definitions.

We mentioned in section

10,

that after

identifying

^two

(or more)

free variables of a

wff,

^{we would}

formally

^{treat the wff as a}

wff with the

original

^{number of} free variables. Under certain cir- cumstances it is

preferable

^{to treat a} wff with identified variables

as a wff with a reduced number of variables. This is done

by

means of a définition. In _case, e.g., ^we want to treat

p(x, x)

^{as a}

wff with a reduced number of

variables,

^{we define:}

The identified variables of _p, that are to be treated as one variable of y, should be mentioned

explicitly

^between the brackets after _p

and q

in the definition.

We will allow a definition of this kind

only

^{in case}

(Ex )p(ae, x)

has been derived

(to

avoid the construction of

empty classes).

A12.0 If

q(x) = dfp(x, x),

^then

~q(x) ~p(x, x).

12.0 If

q(x) =df p(x, x),

^then

q(x)

^and

p(x, x)

^{are inter-}

changeable.

Proof.

As the

only

formal difference between

q(x)

^and

p(x, x) is,

^that

they

^{are to} be treated in different ways when the -- ^or

3-operator

is

applied

^{and as}

it

^{has been}

supposed

^that

(Ex)p(x, x)

^{has been}

derived, they

^are

interchangeable (11.2).

We mentioned in section

11,

that after

replacing

^a free variable

by

^an individual constant, ^{we would}

formally

^{treat the wff as a}

wff with ^a free variable instead of the constant. Under certain cir- cumstances it is

preferable

^{to treat a}

wff,

^after

replacing

^{a free}

variable

by

^{a constant,}

formally

^{as a wff with a} reduced number of variables. This is done

again by

^{means of a} définition. In _case, e.g.,

p(x, c)

has been formed from

p(x, y)

^{and we} want to treat

p(x, c) formally

^{as a wff with one} free variable less than

p(x, y),

we define:

c should be mentioned in the definition

explicitly

between the brackets after _p and not between those after q.

We will allow a definition of this kind

only

^{in case}

has been derived.

If

q(x)

=df

p(x, c),

^then

q(x)

^and

p(x, c)

^{are inter-}

changeable.